*π*-bands of bulk graphene and graphene nanoribbons

^{1,a)}, Mathieu Luisier

^{2}, Gerhard Klimeck

^{2}, Xueping Jiang

^{3}, Neerav Kharche

^{3}, Yu Zhou

^{3}and Saroj K. Nayak

^{3}

### Abstract

Accurate modeling of the *π*-bands of armchair graphenenanoribbons (AGNRs) requires correctly reproducing asymmetries in the bulk graphene bands, as well as providing a realistic model for hydrogen passivation of the edge atoms. The commonly used single-*p _{z} * orbital approach fails on both these counts. To overcome these failures we introduce a nearest-neighbor, three orbital per atom

*p/d*tight-binding model for graphene. The parameters of the model are fit to first-principles density-functional theory –based calculations as well as to those based on the many-body Green’s function and screened-exchange formalism, giving excellent agreement with the

*ab initio*AGNR bands. We employ this model to calculate the current-voltage characteristics of an AGNR MOSFET and the conductance of rough-edge AGNRs, finding significant differences versus the single-

*p*model. These results show that an accurate band structure model is essential for predicting the performance of graphene-based nanodevices.

_{z}This work was partially supported by NSF (Grant No. EEC-0228390) that funds the Network for Computational Nanotechnology; by NSF PetaApps (Grant No. 0749140); by NRI MIND; by NSF through TeraGrid resources provided by the National Institute for Computational Sciences (NICS); and by the Interconnect Focus Center and Computational Center for Nanotechnology Innovations partly funded by New York State (X.J., N.K., Y.Z., and S.K.N.).

I. INTRODUCTION

2. BULK AND HYDROGEN PASSIVATIONMODELS

3. APPLICATION: DEVICE CHARACTERISTICS

a. AGNR-MOSFET model

b. Rough AGNR conductance

IV. CONCLUSIONS

### Key Topics

- Band models
- 38.0
- Graphene
- 26.0
- Density functional theory
- 22.0
- Passivation
- 16.0
- Band gap
- 10.0

## Figures

(a) Bulk bands of graphene from the DFT + GW calculations used to fit the parameters of Table I. The highlighted area around the *K*-point is expanded in part (b). (b) the bulk bands of all three models in the vicinity of *K*: DFT + GW (diamonds), the *p/d* model (solid lines) and the -only model (dotted lines). The -only model has parameters ; the nonzero onsite term is chosen to align its *K* -point with that of the other calculations and the common *K*-point energy is indicated by the heavy dashed horizontal line labeled *E _{K} *. Note the asymmetry of the DFT + GW and

*p/d*bands about

*E*in contrast to the exact symmetry of the -only bands about this energy.

_{K}(a) Bulk bands of graphene from the DFT + GW calculations used to fit the parameters of Table I. The highlighted area around the *K*-point is expanded in part (b). (b) the bulk bands of all three models in the vicinity of *K*: DFT + GW (diamonds), the *p/d* model (solid lines) and the -only model (dotted lines). The -only model has parameters ; the nonzero onsite term is chosen to align its *K* -point with that of the other calculations and the common *K*-point energy is indicated by the heavy dashed horizontal line labeled *E _{K} *. Note the asymmetry of the DFT + GW and

*p/d*bands about

*E*in contrast to the exact symmetry of the -only bands about this energy.

_{K}Gaps (a) and bands (b)–(d) of AGNRs; the DFT and *p/d* model AGNRs are hydrogen passivated while the -only model AGNRs are not, and the uppermost valence band of each model is aligned to facilitate comparison. In panel (a) the DFT gaps are plotted with diamonds, those of the *p/d* model with open circles, and those of the -only model with squares. Lines, which are guides to the eye, denote families: (dashed), (solid), and (dotted). The hydrogen and H-C parameters of Table I were optimized to the DFT results for only AGNR-7, -8, and -9. All other AGNRs use these same parameters. Note the excellent agreement of the *p/d* model with DFT and the poor agreement of the -only results.

Gaps (a) and bands (b)–(d) of AGNRs; the DFT and *p/d* model AGNRs are hydrogen passivated while the -only model AGNRs are not, and the uppermost valence band of each model is aligned to facilitate comparison. In panel (a) the DFT gaps are plotted with diamonds, those of the *p/d* model with open circles, and those of the -only model with squares. Lines, which are guides to the eye, denote families: (dashed), (solid), and (dotted). The hydrogen and H-C parameters of Table I were optimized to the DFT results for only AGNR-7, -8, and -9. All other AGNRs use these same parameters. Note the excellent agreement of the *p/d* model with DFT and the poor agreement of the -only results.

Current-voltage characteristics of the AGNR-12 MOSFET of Ref. 14 as calculated with the -only (dotted lines and closed symbols) and *p/d* models (solid lines and open symbols) for two different drain biases. Computed points are indicated by symbols: circles and squares . Note in Fig. 2(c) the larger AGNR bandgap in the -only model vs the *p/d* model.

Current-voltage characteristics of the AGNR-12 MOSFET of Ref. 14 as calculated with the -only (dotted lines and closed symbols) and *p/d* models (solid lines and open symbols) for two different drain biases. Computed points are indicated by symbols: circles and squares . Note in Fig. 2(c) the larger AGNR bandgap in the -only model vs the *p/d* model.

Differential conductance of a rough AGNR-12 vs line edge roughness probability. Symbols are calculated conductances: Open circles (, *N* channel), solid circles (, *P* channel), open diamonds (*p/d*, *N* channel), or solid diamonds (*p/d*, *P* channel). Each computed point is the average of 250 samples. Note the artificial symmetry of the *N-* and *P*-channel AGNRs in the -only model which does not appear in the *p/d* model.

Differential conductance of a rough AGNR-12 vs line edge roughness probability. Symbols are calculated conductances: Open circles (, *N* channel), solid circles (, *P* channel), open diamonds (*p/d*, *N* channel), or solid diamonds (*p/d*, *P* channel). Each computed point is the average of 250 samples. Note the artificial symmetry of the *N-* and *P*-channel AGNRs in the -only model which does not appear in the *p/d* model.

Differential conductance of a rough AGNR-13 vs line edge roughness probability; note the logarithmic scale. Symbols are calculated conductances: Open circles (, *N* channel), solid circles (, *P* channel), open diamonds (*p/d*, *N* channel), or solid diamonds (*p/d*, *P* channel). Each computed point is the average of 250 samples. Note the artificial symmetry of the *N-* and *P*-channel AGNRs in the -only model which does not appear in the *p/d* model.

Differential conductance of a rough AGNR-13 vs line edge roughness probability; note the logarithmic scale. Symbols are calculated conductances: Open circles (, *N* channel), solid circles (, *P* channel), open diamonds (*p/d*, *N* channel), or solid diamonds (*p/d*, *P* channel). Each computed point is the average of 250 samples. Note the artificial symmetry of the *N-* and *P*-channel AGNRs in the -only model which does not appear in the *p/d* model.

## Tables

C-C and H-C onsite and nearest-neighbor tight-binding parameters used in the *p/d* model; all values are in eV. To simplify the treatment we employ only a single H-C *pd* nearest-neighbor parameter: .

C-C and H-C onsite and nearest-neighbor tight-binding parameters used in the *p/d* model; all values are in eV. To simplify the treatment we employ only a single H-C *pd* nearest-neighbor parameter: .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content